Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian: This is consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe I'm swapping left for right, I guess much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian.

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It would be natural to call a monoid $H$ with the same property hopfian: This would be consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe I'm swapping left for right, I guess much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian; and on the other hand, the term "hopfian monoid" is already taken for something else (see Benjamin Steinberg's comments).

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian: This is consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe it is the other way around, much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian.

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian: This is consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe I'm swapping left for right, I guess much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian.

Is there any standard terminology for a ring in which every right (resp., left) cancellable element is cancellable (or equivalently, every left (resp., right) zero divisor is a zero divisor)? I'm aware of some people going for the term "left (resp., right) eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian (resp., co-hopfian): This is consistent with the abstract definition of a hopfian (resp., co-hopfian) object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe it is the other way around, much depends on the convention adopted for the order of composition of two morphisms). But a hopfian (resp., co-hopfian) ring isn't usually a ring whose multiplicative monoid is hopfian (resp., co-hopfian).

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for the term "left eversible", but the term seems not so well established (and I'm not particularly fond of it).

It is natural to call a monoid $H$ with the same property hopfian: This is consistent with the abstract definition of a hopfian object in a category [Wiki.en], when regarding $H$ as a one-object category in the usual fashion (or maybe it is the other way around, much depends on the convention adopted for the order of composition of two morphisms). But a hopfian ring isn't usually a ring whose multiplicative monoid is hopfian.